The relationship between sound complexity and the underlying morphology and physiology of the vocal organ anatomy is a fundamental component in the evolution of acoustic communication, particularly for fishes. Among vertebrates, the mammalian larynx and avian syrinx are the best-studied vocal organs, and their ability to produce complex vocalizations has been modeled. The range and complexity of the sounds in mammalian lineages have been attributed, in part, to the bilateral nature of the vocal anatomy. Similarly, we hypothesize that the bipartite swim bladder of some species of toadfish (family Batrachoididae) is responsible for complex nonlinear characters of the multiple call types that they can produce, supported by nerve transection experiments. Here, we develop a low-dimensional coupled-oscillator model of the mechanics underlying sound production by the two halves of the swim bladder of the three-spined toadfish, Batrachomoeus trispinosus. Our model was able to replicate the nonlinear structure of both courtship and agonistic sounds. The results provide essential support for the hypothesis that fishes and tetrapods have converged in an evolutionary innovation for complex acoustic signaling, namely, a relatively simple bipartite mechanism dependent on sonic muscles contracting around a gas filled structure.